Pro-P Groups Acting on Trees with Finitely Many Maximal Vertex

Pro-p groups acting on trees with ﬁnitely many maximal vertex stabilizers up to conjugation

Zo´eChatzidakis and Pavel Zalesskii ∗ November 18, 2020

Abstract We prove that a finitely generated pro-p group G acting on a pro-p tree T splits as a free amalgamated pro-p product or a pro-p HNN-extension over an edge stabilizer. If G acts with finitely many vertex stabilizers up to conjugation we show that it is the fundamental pro-p group of a finite graph of pro-p groups (G, Γ) with edge and vertex groups being stabilizers of certain vertices and edges of T respectively. If edge stabilizers are procyclic, we give a bound on Γ in terms of the minimal number of generators of G. We also give a criterion for a pro-p group G to be accessible in terms of the first 1 cohomology H (G, Fp[[G]]).

1 Introduction

The dramatic advance of classical combinatorial group theory happened in the 1970’s, when Bass-Serre theory of groups acting on trees changed completely the face of the theory. The profinite version of Bass-Serre theory was developed by Luis Ribes, Oleg Melnikov and the second author because of the absence of the classical methods of combinatorial group theory for profinite groups. However it does not work in full strength even in the pro-p case. The reason is that if a pro-p group G acts on a pro-p tree T then a maximal subtree of the quotient graph G\T does not always exist and even if it exists it does not always lift to T . As a consequence the pro-p version of Bass-Serre theory does not give subgroup structure theorems the way it does in the classical Bass-Serre theory. In fact, for infinitely generated pro-p subgroups there are counter examples. The objective of this paper is to study the situation when G has only finitely many vertex stabilizers up to conjugation and in this case we can prove the main Bass-Serre theory structure theorem.

Theorem 5.1. Let G be a ﬁnitely generated pro-p group acting on a pro-p tree T with ﬁnitely many maximal vertex stabilisers up to conjugation. Then G is the fundamental group of a

∗The second author partially supported by CNPq and FAPDF.

1 reduced finite graph of finitely generated pro-p groups (G, Γ), where each vertex group G(v) and each edge group G(e) is a maximal vertex stabilizer Gv˜ and an edge stabilizer Ge˜ respectively (for some v,˜ e˜ ∈ T ). In the abstract situation, a finitely generated (abstract) group G acting on a tree has a G-invariant subtree D such that G\D is finite and so has automatically finitely many maximal vertex stabilizers up to conjugation. In the pro-p situation such an invariant subtree does not exist in general and the existence of it in the case of only finitely many stabilizers up to conjugation is not clear even if vertex stabilizers are finite. Nevertheless, for a finitely generated pro-p group acting on a pro-p tree we can prove a splitting theorem into an amalgamated product or an HNN-extension.

Theorem 4.2. Let G be a finitely generated pro-p group acting on a pro-p tree T without global fixed points. Then G splits non-trivially as a free amalgamated pro-p product or pro-p HNN-extension over some stabiliser of an edge of T . This in turn allows us to prove that a non virtually cyclic pro-p group acting on a pro-p tree with finite edge stabilizers has more than one end.

Theorem 4.5. Let G be a finitely generated pro-p group acting on a pro-p tree with finite edge 1 ∼ stabilizers and without global fixed points. Then either G is virtually cyclic and H (G, Fp[[G]]) = 1 Fp (i.e. G has two ends) or H (G, Fp[[G]]) is infinite (i.e. G has infinitely many ends). Theorem 4.2 raises naturally the question of accessibility; namely whether we can continue to split G into an amalgamated free product or HNN-extension forever, or do we reach the situation after finitely many steps where we can not split it anymore. The importance of this is underlined also by the following observation: if a pro-p group G acting on a pro-p tree T is accessible with respect to splitting over edge stabilizers, then by Theorem 4.2 this implies finiteness of the maximal vertex stabilizers up to conjugation and so Theorem 5.1 provides the structure theorem for G. In the abstract situation, accessibility was studied by Dunwoody ([3], [4]) for splitting over finite groups, and by Bestvina and Feighn ([1]) over an arbitrary family of groups. In the pro-p case accessibility was studied by Wilkes ([24]) where a finitely generated not accessible pro-p group was constructed. For a finitely generated pro-p group acting faithfully and irreducibly on a pro-p tree (see Section 2 for definitions) no such example is known. The next theorem gives a sufficient condition of accessibility for a pro-p group; we do not know whether the converse also holds (it holds in the abstract case).

1 Theorem 6.10. Let G be a finitely generated pro-p group. If H (G, Fp[[G]]) is a finitely gen- 1 erated Fp[[G]]-module, then G is accessible. We show here that finitely generated pro-p groups are accessible with respect to cyclic subgroups and in fact give precise bounds.

1proved by G. Wilkes independently in [25].

2 Theorem 6.6. Let G = π1(Γ, G) be the fundamental group of a finite graph of pro-p-groups, with procyclic edge groups, and assume that d(G) = d ≥ 2. Then (assuming that the graph is reduced), the vertex groups are finitely generated, the number of vertices of Γ is ≤ 2d − 1, and the number of edges of Γ is ≤ 3d − 2. Observe that Theorem 6.6 contrasts with the abstract groups situation where for finitely generated groups the result does not hold (see [5]). As a corollary we deduce the bound for pro-p limit groups (pro-p analogs of limit groups introduced in [10], see Section 6 for a precise definition).

Corollary 6.7. Let G be a pro-p limit group. Then G is the fundamental group of a finite graph of finitely generated pro-p groups (G, Γ), where each edge group G(e) is infinite procyclic. Moreover, |V (Γ)| ≤ 2d − 1, and |E(Γ)| ≤ 3d − 2, where d is the minimal number of generators of G. It is worth mentioning that for abstract limit groups the best known estimate for |V (Γ)| is 1 + 4(d(G) − 1), proved by Richard Weidmann in [22, Theorem 1]. In Section 7 we investigate Howson’s property for free products with procyclic amalgamation and HNN-extensions with procyclic associated subgroups. In Section 8.1 we apply the results of Section 7 to normalizers of procyclic subgroups.

Theorem 8.3. Let C be a procyclic pro-p group and G = G1 qC G2 be a free amalgamated pro-p product or a pro-p HNN-extension G = HNN(G1, C, t) of Howson groups. Let U be a procyclic g g subgroup of G and N = NG(U). Assume that NGi (U ) is ﬁnitely generated whenever U ≤ Gi. If K ≤ G is ﬁnitely generated, then so is K ∩ N. Section 2 contains basic notions and facts of the theory of pro-p groups acting on trees used in the paper. The following sections are devoted to the proofs of the results mentioned above. Aknowledgement. This research was started while the second author was visiting the ENS as an invited professor, and the authors thank the ENS for this hospitality.

2 Notation, deﬁnitions and basic results

2.1. Notation. If a pro-p group G continuously acts on a profinite space X we call X a G- space. H1(G) denotes the first homology H1(G, Fp) and is canonically isomorphic to G/Φ(G). −1 g −1 If x ∈ T and g ∈ G, then Ggx = gGxg . We shall use the notation h = g hg for conjugation. If H a subgroup of G, HG will stand for the (topological) normal closure of H in G. If G is an abstract group Gb will mean the pro-p completion of G. 2.2. Conventions. Throughout the paper, unless otherwise stated, groups are pro-p, subgroups will be closed and morphisms will be continuous. Finite graphs of groups will be proper and reduced (see Definitions 2.12 and 2.15). Actions of a pro-p group G on a profinite graph

3 Γ will a priori be supposed to be faithful (i.e., the action has no kernel), unless we consider actions on subgraphs of Γ.

Next we collect basic deﬁnitions, following [17].

2.1 Proﬁnite graphs

Definition 2.3. A profinite graph is a triple (Γ, d0, d1), where Γ is a profinite (i.e. boolean) space and d0, d1 :Γ → Γ are continuous maps such that didj = dj for i, j ∈ {0, 1}. The elements of V (Γ) := d0(G) ∪ d1(G) are called the vertices of Γ and the elements of E(Γ) := Γ \ V (Γ) are called the edges of Γ. If e ∈ E(Γ), then d0(e) and d1(e) are called the initial and terminal vertices of e. A vertex with only one incident edge is called pending. If there is no confusion, one can just write Γ instead of (Γ, d0, d1).

Definition 2.4. A morphism f :Γ → ∆ of graphs is a map f which commutes with the di’s. Thus it will send vertices to vertices, but might send an edge to a vertex.2 Definition 2.5. Every profinite graph Γ can be represented as an inverse limit Γ = lim Γ of ←− i its finite quotient graphs ([17, Proposition 1.5]). A profinite graph Γ is said to be connected if all its finite quotient graphs are connected. Every profinite graph is an abstract graph, but in general a connected profinite graph is not necessarily connected as an abstract graph. 2.6. Collapsing edges. If Γ is a graph and e an edge which is not a loop we can collapse the edge e by removing {e} from the edge set of Γ, and identify d0(e) and d1(e) with a new vertex 0 0 y. I.e., Γ is the graph given by V (Γ ) = V (Γ) \{d0(e), d1(e)} ∪ {y} (where y is the new vertex), 0 0 and E(Γ ) = E(Γ) \{e}. We define π :Γ → Γ by setting π(m) = m if m∈ / {e, d0(e), d1(e)}, 0 0 0 π(e) = π(d0(e)) = π(d1(e)) = y. The maps di :Γ → Γ are defined so that π is a morphism of 0 0 graphs. Another way of describing Γ is that Γ = Γ/∆, where ∆ is the subgraph {e, d0(e), d1(e)} collapsed into the vertex y.

2.2 Pro-p trees 2.7. An exact sequence. Let Γ be a profinite graph, with set of vertices V (Γ) and E(Γ) = Γ \ V (Γ). Let (E∗(Γ), ∗) = (Γ/V (Γ), ∗) be the pointed profinite quotient space with V (Γ) ∗ as distinguished point, and let Fp[[E (Γ), ∗]] and Fp[[V (Γ)]] be respectively the free profinite ∗ Fp-modules over the pointed profinite space (E (Γ), ∗) and over the profinite space V (Γ) (cf. ∗ [15, section 5.2]). Note that when E(Γ) is closed, then Fp[[E (Γ), ∗]] = Fp[[E(Γ)]]. Let the ∗ maps δ : Fp[[E (Γ), ∗]] → Fp[[V (Γ)]] and ε : Fp[[V (Γ)]] → Fp be defined respectively by ∗ δ(e) = d1(e)−d0(e) for all e ∈ E (Γ) and ε(v) = 1 for all v ∈ V (Γ). Then we have the following complex of free profinite Fp-modules

∗ δ ε 0 −−−→ Fp[[E (Γ), ∗]] −−−→ Fp[[V (Γ)]] −−−→ Fp −−−→ 0. 2It is called a quasimorphism in [13].

4 Deﬁnition 2.8. The proﬁnite graph Γ is a pro-p tree if the above sequence is exact. If T is a pro-p tree, then we say that a pro-p group G acts on T if it acts continuously on T and the action commutes with d0 and d1. We say that G acts irreducibly on T if T does not have proper G-invariant subtrees and that it acts faithfully if the kernel of the action is trivial. If t ∈ V (T ) ∪ E(T ) we denote by Gt the stabilizer of t in G. For a pro-p group G acting on a pro-p tree T we let G˜ denote the subgroup generated by all vertex stabilizers. Moreover, for any two vertices v and w of T we let [v, w] denote the geodesic connecting v to w in T , i.e., the (unique) smallest pro-p subtree of T that contains v and w.

2.3 Finite graphs of pro-p groups When we say that G is a finite graph of pro-p groups we mean that it contains the data of the underlying finite graph, the edge pro-p groups, the vertex pro-p groups and the attaching continuous maps. More precisely, Definition 2.9. let Γ be a connected finite graph. A graph of pro-p groups (G, Γ) over Γ consists S of specifying a pro-p group G(m) for each m ∈ Γ (i.e. G = · m∈Γ G(m)), and continuous monomorphisms ∂i : G(e) −→ G(di(e)) for each edge e ∈ E(Γ), i = 1, 2. Definition 2.10. (1)A morphism of graphs of pro-p groups: (G, Γ) → (H, ∆) is a pair (α, α¯) of maps, with α : G −→ H a continuous map, andα ¯ :Γ −→ ∆ a morphism of graphs, and such that αG(m) : G(m) −→ H(¯α(m)) is a homomorphism for each m ∈ Γ and which commutes with the appropriate ∂i. Thus the diagram

α G / H

∂i ∂i α G / H is commutative.

(2) We say that (α, α¯) is a monomorphism if both α, α¯ are injective. In this case its image will be called a subgraph of groups of (H, ∆). In other words, a subgraph of groups of a graph of pro-p-groups (G, Γ) is a graph of groups (H, ∆), where ∆ is a subgraph of Γ (i.e., E(∆) ⊆ E(Γ) and V (∆) ⊆ V (Γ), the maps di on ∆ are the restrictions of the maps di on Γ), and for each m ∈ ∆, H(m) ≤ G(m). 2.11. Deﬁnition of the fundamental group. The pro-p fundamental group

G = Π1(G, Γ)

of the graph of pro-p groups (G, Γ) is deﬁned by means of a universal property: G is a pro-p group together with the following data and conditions: (i) a maximal subtree D of Γ;

5 (ii) a collection of continuous homomorphisms

νm : G(m) −→ G (m ∈ Γ),

and a continuous map E(Γ) −→ G, denoted e 7→ te (e ∈ E(Γ)), such that te = 1 if e ∈ E(D), and

−1 (νd0(e)∂0)(x) = te(νd1(e)∂1)(x)te , ∀x ∈ G(e), e ∈ E(Γ);

(iii) the following universal property is satisﬁed:

whenever one has the following data

• H is a pro-p group,

• βm : G(m) −→ H,(m ∈ Γ), a collection of continuous homomorphisms,

• a map e 7→ se (e ∈ E(Γ)) with se = 1 if e ∈ E(D), and

−1 • (βd0(e)∂0)(x) = se(βd1(e)∂1)(x)se , ∀x ∈ G(e), e ∈ E(Γ),

then there exists a unique continuous homomorphism δ : G −→ H such that δ(te) = se (e ∈ E(Γ)), and for each m ∈ Γ the diagram

G < ν yy m yy yy yy G(m) δ E EE EE βm EE E " H

commutes.

The main examples of Π1(G, Γ) are an amalgamated free pro-p product G1 qH G2 and an HNN-extension HNN(G, H, t) that correspond to the case of Γ having one edge and two and one vertex respectively.

Deﬁnition 2.12. We call the graph of groups (G, Γ) proper (injective in the terminology of [13]) if the natural map G(v) → Π1(G, Γ) is an embedding for all v ∈ V (Γ).

6 Remark 2.13. In the pro-p case, a graph of groups (G, Γ) is not always proper. However, the vertex and edge groups can always be replaced by their images in Π1(G, Γ) so that (G, Γ) becomes proper and Π1(G, Γ) does not change. Thus throughout the paper we shall only consider proper graphs of pro-p groups. In particular, all our free amalgamated pro-p products are proper. If (G, Γ) is a finite graph of finitely generated pro-p groups, then by a theorem of J-P. Serre (stating that every finite index subgroup of a finitely generated pro-p group is open, cf. [15, §4.8]) the fundamental pro-p group G = Π1(G, Γ) of (G, Γ) is the pro-p completion of the usual fundamental group π1(G, Γ) (cf. [19, §5.1]). Note that (G, Γ) is proper if and only if π1(G, Γ) is residually p. In particular, edge and vertex groups will be subgroups of Π1(G, Γ). 2.14. Presentation of the fundamental group. In [29, paragraph (3.3)], the fundamental group G is defined explicitly in terms of generators and relations associated to a chosen subtree D. Namely

−1 G = hG(v), te | v ∈ V (Γ), e ∈ E(Γ), te = 1 for e ∈ D, ∂0(g) = te∂1(g)te , for g ∈ G(e)i (1) I.e., if one takes the abstract fundamental group G = π (G, Γ), then Π (G, Γ) = lim G /N, 0 1 1 ←−N 0 where N ranges over all normal subgroups of G0 of index a power of p and with N ∩ G(v) open in G(v) for all v ∈ V (Γ). Note that this last condition is automatic if G(v) is finitely generated (as a pro-p-group). It is also proved in [29] that the definition given above is independent of the choice of the maximal subtree D. Definition 2.15. A finite graph of pro-p groups (G, Γ) is said to be reduced, if for every edge e which is not a loop, neither ∂1(e): G(e) → G(d1(e)) nor ∂0(e): G(e) → G(d0(e)) is an isomorphism. Remark 2.16. Any finite graph of pro-p groups can be transformed into a reduced finite graph of pro-p groups by the following procedure: If {e} is an edge which is not a loop and for which 0 one of ∂0, ∂1 is an isomorphism, we can collapse {e} to a vertex y (as explained in 2.6). Let Γ 0 0 be the finite graph given by V (Γ ) = {y} ∪· V (Γ) \{d0(e), d1(e)} and E(Γ ) = E(Γ) \{e}, and 0 0 0 0 let (G , Γ ) denote the finite graph of groups based on Γ given by G (y) = G(d1(e)) if ∂0(e) is 0 an isomorphism, and G (y) = G(d0(e)) if ∂0(e) is not an isomorphism.

This procedure can be continued until ∂0(e), ∂1(e) are not surjective for all edges not defin- ing loops. Note that the reduction process does not change the fundamental pro-p group, i.e., one has a canonical isomorphism Π1(G, Γ) ' Π1(Gred, Γred). So, if the pro-p group G is the fundamental group of a finite graph of pro-p groups, we may assume that the finite graph of pro-p groups is reduced.

2.17. Standard (universal) pro-p tree. Associated with the ﬁnite graph of pro-p groups (G, Γ) there is a corresponding standard pro-p tree (or universal covering graph) T = T (G) =

7 S · m∈Γ G/G(m) (cf. [29, Proposition 3.8]). The vertices of T are those cosets of the form gG(v), with v ∈ V (Γ) and g ∈ G; its edges are the cosets of the form gG(e), with e ∈ E(Γ); and the incidence maps of T are given by the formulas:

d0(gG(e)) = gG(d0(e)); d1(gG(e)) = gteG(d1(e)) (e ∈ E(Γ), te = 1 if e ∈ D). There is a natural continuous action of G on T , and clearly G\T = Γ. There is a standard connected transversal s :Γ → T , given by m 7→ G(m). Note that s|D is an isomorphism of graphs and the elements te satisfy the equality d1(s(e)) = tes(d1(e)). Using the map s, we shall identify G(m) with the stabilizer Gs(m) for m ∈ Γ:

−1 G(e) = Gs(e) = Gd0(s(e)) ∩ Gd1(s(e)) = G(d0(e)) ∩ teG(d1(e))te (2)

with te = 1 if e ∈ D. Remark also that since Γ is finite, E(T ) is compact. 2.18. The fundamental group of a profinite graph. If all vertex and edge groups are trivial we get the definition of the pro-p fundamental group π1(Γ). It follows that π1(Γ) is a free abs pro-p group on the base Γ\D and so coincides with the pro-p completion πb1 (Γ) of the abstract abs (usual) fundamental group π1 (Γ) that also can be defined traditionally by closed circuits. Therefore if Γ is connected profinite and Γ = lim Γ is an inverse limit of finite graphs it induces ←− i the inverse system {π (Γ ) = πabs(Γ )} and π (Γ) is defined as π (Γ) = lim π (Γ ) in this 1 i b1 i 1 1 ←−i 1 i case. The fundamental group π1(Γ) acts freely on a pro-p tree Γe (universal cover) such that ˜ π1(Γ)\Γ = Γ (see [27] or [13, Chapter 3] for details). We shall use frequently in the paper the following known results. Proposition 2.19. ([17, Lemma 3.11]). Let G be a pro-p group acting on a pro-p tree T . Then there exists a non-empty minimal G-invariant subtree D of T . Moreover, if G does not stabilize a vertex, then D is unique. Theorem 2.20. ([17, Theorem 3.9]) Let G be a finite p-group acting on a pro-p tree T . Then G fixes a vertex of T . Theorem 2.21. ([9, Proposition 2.4] or [13, Theorem 9.6.1]) Let G be a pro-p group acting on a second countable (as a topological space) pro-p tree T with trivial edge stabilizers. Then there exist a continuous section σ : G\V (T ) −→ V (T ) and a G = Gσ(v) q F, v∈G\V (T ) where F is a free pro-p group naturally isomorphic to G/G˜.

Theorem 2.22. ([13, Theorem 7.1.2], [29, Theorem 3.10]) Let G = Π1(G, Γ) be the fundamental pro-p group of a finite graph of pro-p groups (G, Γ). Then any finite subgroup K of G is conjugate into some vertex group G(v). In particular, if the groups G(v) are finite, they are exactly the maximal finite subgroups of G up to conjugation.

8 We ﬁnish this section with a proposition that completes a missing case of [23, Corollary 3.3], and which is needed in our paper as well as in [23].

Proposition 2.23. Let (G, Γ) be a reduced finite graph of finite p-groups, and suppose that G = Π1(G, Γ, v0) contains a free open subgroup H. Then there exist finitely many reduced finite 0 0 0 0 graphs of finite p-groups (G , Γ ) up to isomorphism such that G ' Π1(G , Γ , w0). Proof. The case of non-procyclic H was proved in [23, Corollary 3.3]. If H is trivial, then G is finite and so the reduced graph of groups (G, Γ) is a vertex v with G = G(v). So we assume that H 6= 1 is procyclic. Then G acts irreducibly on the standard pro-p tree T (G) (see 2.17) and the kernel K of the action has order less than or equal to [G : H]. Hence G/K acts faithfully ∼ on T (G) and so by [17, Theorem 3.15] either G/K = Zp or G/K = C2 q C2 is infinite dihedral. In the first case G = K o Zp = HNN(K, K, t) and in the second case G = G1 qK G2, with [G1 : K] = 2 = [G2 : K]. Note that K is the unique finite normal subgroup of G, and the result follows easily, using Theorem 2.22.

3 Preliminaries: Auxiliary results

In this section we shall prove several auxiliary results on proﬁnite graphs and pro-p groups acting on trees and which are needed later in the paper.

Lemma 3.1. (cf. [19, Corollary 2]) Let ν : ∆ −→ Γ be a morphism of ﬁnite connected graphs representing the collapse of an edge, not a loop. If T is a maximal subtree of Γ, then ν−1(T ) is a maximal subtree of ∆.

Proof. Consider ν−1(T ) . Since V (Γ) ⊂ T , V (∆) ⊂ ν−1(T ). Since ν−1(T ) contains a collapsed edge, |E(ν−1(T ))| = |E(T )| + 1 and ν−1(T ) is connected. Thus |E(ν−1(T ))| = |E(T )| + 1 = |V (Γ)| = |V (∆)| − 1. Since ν−1(T ) is connected, it must be a tree, as needed.

Lemma 3.2. Let Γ be a proﬁnite graph and ∆ an abstract connected subgraph of ﬁnite diameter n (i.e. the shortest path between any two vertices has length at most n). Then the closure ∆ of ∆ in Γ has diameter at most n.

Proof. Write Γ = lim Γ as an inverse limit of finite quotient graphs and let ∆ be the image ←− i i of ∆ in Γ . Then ∆ is finite and has diameter not more than n. Since ∆ = lim ∆ , so does i i ←− i ∆. Indeed, pick two vertices v, w in ∆ and let vi, wi their images in ∆i. The set Ωi of paths of length at most n between v and w is finite and non-empty. Then Ω = lim Ω consists of paths i i ←− i between v and w of length not greater than n and is non-empty.

Proposition 3.3. Let Γ be a connected profinite graph of finite diameter. If π1(Γ) is finitely generated, then there exists a finite connected subgraph ∆ of Γ such that π1(Γ) = π1(∆).

9 Proof. By [8, Corollary 4] Γ is connected as an abstract graph. Then by [21, Proposition 2.7] abs abs π1(Γ) is the pro-p completion of the usual fundamental group π1 (Γ) and so π1 (Γ) is a free group of the same rank n as π1(Γ). Let D be an abstract maximal subtree of the abstract graph Γ. Then |Γ \ D| < ∞. Let e1, . . . , en be all edges from Γ \ D. Then π1(Γ)) is free pro-p of rank n. Let Ω be a minimal subtree of D containing all vertices of e1, . . . , en. Since Γ has finite diameter, Ω is finite. Therefore ∆ = Ω ∪ {e1 ∪ · · · ∪ en} is a finite connected subgraph abs of Γ and π1 (∆) is a free group of rank n. But the fundamental group of a subgraph is a free abs abs factor of the fundamental group of a graph, so π1 (∆) = π1 (Γ) and so their pro-p completions π1(∆) = π1(Γ).

˜ Proposition 3.4. Let G be a pro-p group acting on a pro-p tree T . Then G/G = π1(G\T ) is ˜ a free pro-p group acting freely on G\T . Moreover, if G\T is ﬁnite, then the rank of π1(G\T ) is |E(G\T )| − |V (G\T )| + 1. ˜ Proof. Recall that G is the closed subgroup of G generated by the vertex stabilisers Gv, v ∈ T . ˜ By [13, Corollary 3.9.3] G/G = π1(G\T ) is free pro-p and by Proposition [17, Proposition 3.5] G˜\T is a pro-p tree. If Γ := G\T is ﬁnite it has a maximal subtree D and by [13, Theorem 3.7.4] a basis of π1(G\T ) is Γ \ D. Since V (D) = V (G\T ) the result follows.

Proposition 3.5. Let G be a finitely generated pro-p group acting on a pro-p tree T such that Γ = G\T has finite diameter. Then T possesses a G-invariant subtree D such that G\D is finite. ˜ Proof. By Proposition 3.4, G = G o π1(Γ). We first show that there are finitely many vertices

w1, . . . , wn such that G = hGwi , π1(Γ) | i = 1, . . . , ni. Indeed, let f : G −→ G/Φ(G) be the natural epimorphism to the quotient modulo the ˜ ˜ Frattini subgroup. Then G/Φ(G) = f(G) ⊕ f(π1(Γ)) and since f(G) is ﬁnite (as G/Φ(G) ˜ is) there are vertices w1, . . . , wn of T such that f(G) = hf(Gw1 ), . . . , f(Gwn )i. Hence G =

hGwi , π1(Γ) | i = 1, . . . , ni. Now since G is finitely generated, so is π1(Γ) and therefore by Proposition 3.3, Γ contains a finite subgraph ∆ such that π1(∆) = π1(Γ). Let v1, . . . , vn be the images of w1, . . . , wn in Γ and Ω a minimal connected graph containing ∆ and v1, . . . , vn. Clearly (because Γ has finite diameter) 0 0 Ω is finite and so there exists a connected transversal Σ of Ω in T . Let w1, . . . , wn be the vertices 0 of Σ whose images in Ω are v1, . . . vn respectively. Since for each i we have wi = giwi for some g ∈ G and so G 0 is a conjugate of G in G, it follows that G = hG 0 , π (Γ) | i = 1, . . . ni. Let i wi wi wi 1 D be the connected component of the inverse image of Ω in T containing Σ. We show that D is G-invariant. Let H = Stab(D) be the setwise stabilizer of D in G. Clearly, G 0 ≤ H for each i. wi By [2, Lemma 2.14], we have H\D = Ω. Note that ∆ ⊆ Ω ⊆ Γ and so π1(∆) = π1(Ω) = π1(Γ). ˜ By Proposition 3.4 H = H o π1(Ω), i.e. we may assume that π1(Γ) = π1(Ω) is contained in H. But then G = H and G\D = Ω is finite as desired.

10 Lemma 3.6. Let G = Π1(G, Γ) be the fundamental group of a ﬁnite graph of pro-p groups (G, Γ). Let D be a maximal subtree of Γ, n be the number of pending vertices of D. Then n ≤ 3d(G), where d(G) is the minimal number of generators of G, and |Γ \ D| ≤ d(G). Proof. For every pending vertex v of Γ and the (unique) edge e ∈ D connected to it, G(v) = G(v)/G(e)G(v) is non-trivial, because the graph of groups (G, Γ) is reduced, and the groups are pro-p. Deﬁne the quotient graph of groups (G, Γ) by putting G(m) = 1 if m ∈ Γ is not a pending vertex and G(v) = G(v)/G(e)G(v) 6= 1 if v is a pending vertex. Then from the presentation (1) ` for Π1(G, Γ) it follows then that Π1(G, Γ) = v∈V (Γ) G(v) q π1(Γ). The natural morphism (G, Γ) −→ (G, Γ) induces then the epimorphism G = Π1(G, Γ) −→ G = Π1(G, Γ). The number of pending vertices of Γ is at most d(G) ≤ d(G). On the other hand the rank of π1(Γ) equals |Γ \ D| and is not greater than d(G) by Proposition 3.4. Every edge of Γ \ D connects at most two pending vertices of D and so the number of pending vertices of D is at most 3d(G) ≤ 3d(G).

Lemma 3.7. Let G be a pro-p group acting on a pro-p tree T with |G\T | < ∞. Let H be a subgroup of G with an H-invariant subtree D of T such that the natural map H\D −→ G\T is injective. Then G = Π1(G,G\T ), H = Π1(H,H\D) and (H,H\D) is a subgraph of groups of (G,G\T ). Proof. A maximal subtree of H\D can be extended to a maximal subtree of G\T and so we can choose a connected transversal S of H\D in D that extends to a connected transversal Σ of G\T in T . We may further suppose that if an edge e is in S or Σ, then so is d0(e). Let ρ : T −→ G\T be the natural epimorphism. Then we can deﬁne the graph of groups (H,H\D) and (G,G\T ) in the standard manner,

as follows. If s ∈ S, deﬁne H(ρ(s)) = Hs; if e ∈ S is an edge, deﬁne ∂0 : Hρ(e) → Hρ(d0(e)) to

be the natural inclusion He → Hd0(e), and if ked1(e) ∈ S, deﬁne ∂1 : Hρ(e) → Hρ(d1(e)) to be −1 the natural inclusion He → Hd1(e) followed by conjugation by ke : Hρ(d1(e)) → Hρ(ked1(e)). The deﬁnition is similar for (G,G\T ). By [13, Proposition 3.10.4 and Theorem 6.6.1], we then have G = Π1(G,G\T ), H = Π1(H,H\D).

4 Splitting of pro-p groups acting on trees

Lemma 4.1. Let G be a finitely generated pro-p group acting on a pro-p tree T . Then G = ˜ ˜ lim G/U and G/U = Π1(GU , ΓU ) is the fundamental group of a finite reduced graph of ←−U/oG ˜ finite p-groups. Moreover, the inverse system {G/U, πVU } can be chosen in such a way that it is linearly ordered and for each {G/V˜ } of the system with V ≤ U there exists a natural morphism (ηVU , νVU ):(GV , ΓV ) −→ (GU , ΓU ) where νVU is just a collapse of edges of ΓV and ηVU (GV (m)) = πVU (GV (m)); the induced homomorphism of the pro-p fundamental groups ˜ ˜ coincides with the canonical projection πVU : G/V −→ G/U.

11 ˜ Proof. Recall that U is the closed subgroup of G generated by the vertex stabilizers Uv. Clearly ˜ ˜ ˜ ˜ ˜ G/U and U/U act on U\T ; by Proposition 3.4 U/U is free pro-p. Thus GU := G/U is virtually free pro-p. By [7, Theorem 1.1] it follows that GU is the fundamental pro-p group Π1(GU , ΓU ) of a finite graph of finite p-groups. As mentioned in Section 2 we may assume that (GU , ΓU ) is reduced. Although the finite graph of finite p-groups (GU , ΓU ) is not uniquely determined by U, the index U in the notation shall express that these objects are depending on U. Since the maximal finite subgroups of GU are exactly the vertex groups of (GU , ΓU ) up to conjugation (see Theorem 2.22), the number of vertices of ΓU does not depend on the choice of (GU , ΓU ), ˜ and since π1(ΓU ) = U/U is free pro-p of rank |E(ΓU )| − |V (ΓU )| + 1, the size of ΓU is bounded in terms of possible decompositions as a reduced finite graph of finite p-groups of U/U˜. Clearly we have G = lim G . ←−U U By [18, Prop. 1.10], viewing GU as a quotient of GV when V ≤ U (via the natural map ˜ ˜ ˜ πVU : G/V −→ G/U), one has a natural decomposition of G/U as the pro-p fundamental ˜ group G/U = Π1(GVU , ΓV ) of a finite graph of finite p-groups (GVU , ΓV ), where the vertex and edge groups satisfy GVU (x) = πVU (GV (x)), x ∈ V (ΓV ) ∪· E(ΓV ). Thus we have a morphism ηVU :(GV , ΓV ) −→ (GVU , ΓV ) of graphs of groups such that the induced homomorphism on the pro-p fundamental groups coincides with the canonical projection πVU .

If (GVU , ΓV ) is not reduced, then collapsing some fictitious edges ei, i = 1, . . . , k, we arrive at a reduced graph of groups ((GVU )red, ∆V ). By Proposition 2.23, the number of isomorphism classes of finite reduced graphs of finite p-groups (G0, ∆) which are based on a finite graph ∆ ˜ 0 and satisfy G/U ' Π1(G , ∆) is finite. Using this remark, for each open normal subgroup V we let ΩV be the (finite) set of reduced ˜ finite graphs of finite p-groups (GV , ΓV ) with G/V ' Π1(GV , ΓV ). Let Vi, i ∈ N, be a decreasing T chain of open normal subgroups of G with V0 = U and i Vi = (1). For X ⊆ ΩVi define T (X) to

be the set of all reduced graphs of groups in ΩVi−1 that can be obtained from graphs of groups in X by the procedure explained in the preceding paragraph (note that T does not define a (i) map on X). Define Ω1 = T (ΩV1 ), Ω2 = T (T (ΩV2 )), . . . , Ωi = T (ΩVi ) and note that Ωi is a non-empty subset of ΩU for every i ∈ N. Clearly Ωi+1 ⊆ Ωi and since ΩU is finite there is an

i1 ∈ N such that Ωj = Ωi1 for all j > i1 and we denote this Ωi1 by ΣU . Then T (ΣVi ) = ΣVi−1

for all i, and so we can construct an inﬁnite sequence of graphs of groups (GVj , Γj) ∈ ΩVj

such that (GVj−1 , Γj−1) ∈ T (GVj , Γj) for all j. This means that (GVj Vj−1 , ΓVj ) can be reduced

to (GVj−1 , Γj−1), i.e., that this sequence {(GVj , Γj)} is an inverse system of reduced graphs of groups satisfying the required conditions.

Note that in the classical Bass-Serre theory, a finitely generated group G acting irreducibly on a tree T has finitely many orbits, i.e. G\T is finite. This is not the case in the pro-p case; this fact highlights the complementary difficulties that appear in the pro-p case. The next result partially overcomes this.

Theorem 4.2. Let G be a ﬁnitely generated pro-p group acting on a pro-p tree T without global ﬁxed points. Then G splits non-trivially as a free amalgamated pro-p product or pro-p

12 HNN-extension over some stabilizer of an edge of T . ˜ ˜ Proof. By Lemma 4.1 G = lim G/U, where G/U = Π1(GU , ΓU ) is the fundamental group ←−U/oG of a finite reduced graph of finite p-groups and for each V/o G contained in U, one has a natural morphism (ηVU , νVU ):(GV , ΓV ) −→ (GU , ΓU ) such that νVU is just a collapse of edges of ΓV . Moreover, the induced homomorphism of the pro-p fundamental groups coincides with ˜ ˜ the canonical projection πVU : G/V −→ G/U. Note that U/U˜ is non-trivial for some U, since otherwise G/U˜ is finite for every U and by Theorem 2.20 stabilizes a vertex vU ; hence by an inverse limit argument G would stabilize a vertex v in T contradicting the hypothesis. Hence ΓU contains at least one edge.

Case 1. There exists U and an edge eU in ΓU such that ΓU \{eU } is disconnected. Let eV be an edge of ΓV such that νVU (eV ) = eU . Since ΓU is obtained from ΓV by collapsing

edges, ΓV \{eV } is disconnected as well. Thus we may write GV = AV qGV (eV ) BV , where AV , BV are the fundamental groups of the graphs of groups (GV , ΓV ) restricted to the connected components of ΓV \{eV }, and we have an inverse limit of free amalgamated products that gives a decomposition G = A q B for some e ∈ E(T ), with A = lim A , B = lim B . G(e) ←−V V ←−V V Case 2. For all U and each edge eU of ΓU the graph ΓU \{eU } is connected. By Lemma 3.1 (applied inductively) the preimage in ΓV of a maximal subtree DU of ΓU is a maximal subtree DV of ΓV . Therefore for each V we have ˜ G/V = HNN(LV , GV (e), te, e ∈ ΓV \ DV ),

˜ ˜ ˜ where LV = Π1(GV ,DV ). Note that the image of G in G/V is G/]V and since G is ﬁnitely ˜ generated, so is G/G. Therefore, by Proposition 3.4, π1(ΓV ) = F (ΓV \ DV ) is a free pro- ˜ p group of rank |ΓV \ DV | ≤ rank(G/G), i.e. we can assume that ΓV \ DV is a constant set E. Then we can view E as a ﬁnite subset of E(T ) and putting L = lim L we have ←−V V G = HNN(L, Ge, te, e ∈ E) for some e ∈ E(T ) as required.

Remark 4.3. By Lemma 4.1, a finitely generated pro-p group G acting on a pro-p-tree T can be represented as an inverse limit of fundamental groups of finite reduced graphs of pro-p-groups. There is however no bound on the size of the finite graphs. The next result will show that given any finite graph ΓU occuring in this inverse limit, G admits a representation as Π1(G, ΓU ). Corollary 4.4. Let G be a finitely generated pro-p-group G which acts on a pro-p tree T , and let U, GU , ΓU , GU satisfy the conclusion of Lemma 4.1. Then G splits as the pro-p fundamental group of a reduced finite graph of pro-p groups G = Π1(G, ΓU ) with edge groups being stabilizers of some edges of T .

Proof. We use induction on the size of ΓU . If |ΓU | = 1, there is nothing to prove. If |ΓU | > 1, ˜ ˜ then G/U is inﬁnite. Let πU : G −→ G/U be the natural projection. Pick eU ∈ E(ΓU ). If ΓU \{eU } = ∆ ∪· Ω is disconnected with two connected components ∆ and Ω, then from the proof of Theorem 4.2 it follows that G splits as an amalgamated free product A qGe B

13 with πU (Ge) = GU (eU ), where πU (A) and πU (B) are the fundamental groups of the reduced graphs of groups (GU , ∆) and (GU , Ω) that are the restrictions of (GU , ΓU ) to these connected components. Moreover, assuming w.l.o.g d0(e) ∈ ∆, d1(e) ∈ Ω we have embeddings ∂0 : GU (e) −→ GU (d0(e)) ≤ πU (A), ∂1 : GU (e) −→ GU (d1(e)) ≤ πU (B) for each U that induce the natural embeddings G(e) −→ G(d0(e)) ≤ A, G(e) −→ G(d1(e)) ≤ B. Hence from the induction hypothesis A = Π1(G, ∆), B = Π1(G, Ω) and the result follows. If ΓU \{eU } is connected then again from the proof of Theorem 4.2 it follows that GU splits as an HNN-extension GU = HNN(L, GU (e), te, e ∈ ΓU \ DU ), where DU is a maximal subtree of ΓU \{eU } and πU (Ge) = GU (eU ), πU (L) = π1(GU ,DU ). Moreover, we have embeddings ∂i : GU (e) −→ GU (di(e)) ≤ πU (L), for each U that induce the natural embeddings G(e) −→ G(di(e)) ≤ L. Then by induction hypothesis L = Π1(G,DU ) and G = Π1(G, ΓU ) as needed. Finally we observe that (G, ΓU ) is reduced since (GU , ΓU ) is.

A.A. Korenev [11] defined the number of pro-p ends e(G) for an infinite pro-p group G as 1 e(G) = 1 + dimH (G, Fp[[G]]). The next theorem shows that similar to the abstract case a pro-p group acting irreducibly on an infinite pro-p tree with finite edge stabilizers has more than one end.

Theorem 4.5. Let G be a finitely generated pro-p group acting on a pro-p tree T with finite edge 1 ∼ stabilizers and without global fixed points. Then either G is virtually cyclic and H (G, Fp[[G]]) = 1 Fp (i.e. G has two ends) or H (G, Fp[[G]]) is infinite (i.e. G has infinitely many ends). Proof. By Theorem 4.2, G splits either as an amalgamated free pro-p product or an HNN- extension over an edge stabilizer Ge and so acts on the standard pro-p tree T (G) associated with this splitting. Let H be an open normal subgroup of G intersecting Ge trivially. Then H acts on T (G) with trivial edge stabilizers and so by Theorem 2.21 H is a non-trivial free pro-p product H = H1 q H2. Then we have the following exact sequence (associated to the standard pro-p tree) for this free product decomposition:

δ ε 0 → Fp[[H]] −→ Fp[[H/H1]] ⊕ Fp[[H/H2]] −→ Fp → 0 (∗)

Claim. The augmentation ideal I(H) is decomposable as an Fp[[H]]-module. Proof. Let M1 and M2 be the kernels of the restrictions of ε to Fp[[H/H1]] and Fp[[H/H2]] respectively. We will show that δ(I(H)) = M1 ⊕ M2. Since δ(Fp[[H]]) = ker(ε), M1 ⊕ M2 is a submodule of δ(Fp[[H]]) and since the middle term of (∗) modulo M1 ⊕ M2 is Fp ⊕ Fp, it is of index p in ker(ε). But Fp[[H]] is a local ring and so has a unique maximal left ideal, hence δ(I(H)) = M1 ⊕ M2 as needed. The claim is proved.

Now applying HomFp[[H]](−, Fp[[H]]) to

0 → I(H) → Fp[[H]] → Fp → 0

14 and observing that by [11, Lemma 3]

H HomFp[[H]](Fp, Fp[[H]]) = (Fp[[H]]) = 0, HomFp[[H]](Fp[[H]], Fp[[H]]) = Fp[[H]] and Ext1 ( [[H]],M) = 0 since [[H]] is a free pro-p module, we obtain the exact sequence Fp[[H]] Fp Fp

ϕ 1 0 → Fp[[H]] −→ HomFp[[H]](I(H), Fp[[H]]) → H (H, Fp[[H]]) → 0. (Here we also use that Ext1 ( ,M) = H1(H,M) for an [[H]]-module M). Since Fp[[H]] Fp Fp Fp[[H]] is indecomposable and ∼ HomFp[[H]](I(H), Fp[[H]]) = HomFp[[H]](M1, Fp[[H]]) ⊕ HomFp[[H]](M2, Fp[[H]])

1 (from the claim), ϕ is not onto and so H (H, Fp[[H]]) 6= 0. 1 Then by [11, Theorems 1 and 2], the dimension of H (H, Fp[[H]]) is either inﬁnite or 1 and 1 ∼ 1 in the latter case H is virtually cyclic. By [11, Lemma 2], H (H, Fp[[H]]) = H (G, Fp[[G]]), hence the result.

5 Subgroups of fundamental groups of graphs of pro-p groups

In the classical Bass-Serre theory of groups acting on trees a finitely generated group G acting on a tree T is the fundamental group of a finite graph of groups whose edge and vertex groups are G-stabilizers of edges and vertices of T respectively. This is due to the fact that for finitely generated G there exists a G-invariant subtree D such that G\D is finite. In the pro-p situation this is not always the case. Note that G\D finite implies that there are only finitely many maximal stabilizers of vertices of T in G up to conjugation. In this section we prove the above mentioned result in the pro-p case under the assumption of finitely many maximal vertex stabilizers up to conjugation.

Theorem 5.1. Let G be a finitely generated pro-p group acting on a pro-p tree T with finitely many maximal vertex stabilizers up to conjugation. Then G is the fundamental group of a reduced finite graph of pro-p groups (G, Γ), where each vertex group G(v) and each edge group G(e) is a maximal vertex stabilizer Gv˜ and an edge stabilizer Ge˜ respectively (for some v,˜ e˜ ∈ T ). ˜ ˜ Proof. By Lemma 4.1, G = lim G/U, where G/U = Π1(GU , ΓU ) is the fundamental group ←−U/oG of a finite reduced graph of finite p-groups and for each V/0 G contained in U one has a natural morphism (ηVU , νVU ):(GV , ΓV ) −→ (GU , ΓU ) such that ν is just a collapse of edges of ΓV . Moreover, the induced homomorphism of the pro-p fundamental groups coincides with the ˜ ˜ canonical projection πVU : G/V −→ G/U. We claim now that the number of vertices and edges of ΓU is bounded independently of

U. Let Gv1 ,...,Gvn be the maximal vertex stabilizers of G up to conjugation. Then GU and

15 U/U˜ act on U˜\T ; by Proposition 3.4, the quotient group U/U˜ acts freely on the pro-p tree U˜\T . Thus all vertex stabilizers of G/U˜ are finite and are the images of the corresponding vertex stabilizers of G. Note that any finite subgroup of G/U˜ stabilizes a vertex (Theorem 2.20) and since the maximal finite subgroups of GU are exactly the vertex groups of (GU , ΓU ) up to conjugation (see Theorem 2.22), we see that the number of vertices of ΓU is bounded by n. ∼ ˜ Since π1(ΓU ) = G/G is free pro-p of rank |E(ΓU )| − |V (ΓU )| + 1 by Proposition 3.4, the number of edges of ΓU is bounded by n + d(G) − 1 and so the size of ΓU is bounded independently of U. Hence, for some U and all V ≤ U, the maps νVU :ΓV → ΓU are isomorphisms, and we will denote this graph by Γ. Then for (G, Γ) = lim (G , Γ) we have G(x) = lim G (x) if x is either a vertex or an edge of ←− U ←− U Γ, and (G, Γ) is a reduced finite graph of pro-p groups satisfying G ' Π1(G, Γ). This finishes the proof of the theorem.

Corollary 5.2. The number up to conjugation of maximal vertex stabilizers in G equals |V (Γ)|.

Proof. Since (G, Γ) = lim (G , Γ), the result follows from Theorem 5.1. ←−U U

One of the main consequences of the main theorem of Bass-Serre theory is an extension of the Kurosh subgroup theorem to a group G acting on a tree T . Namely if H is a subgroup of G then H = π1(H, ∆) is the fundamental group of a graph of groups constructed as follows. Let ∆ = H\T and if Σ is a connected transversal of ∆ in T then H consists of the stabilizers of the edges and vertices of Σ. In the pro-p situation such a theorem does not hold in general ([6, Theorem 1.2]). Our next objective is to prove it for H acting acylindrically and having ﬁnitely many maximal vertex stabilizers up to conjugation.

Deﬁnition 5.3. The action of a pro-p group G on a pro-p tree T is said to be k-acylindrical, for k a constant, if for every g 6= 1 in G, the subtree T g of ﬁxed points has diameter at most k.

Theorem 5.4. Let G be a ﬁnitely generated pro-p group acting n-acylindrically on a pro-p tree T with ﬁnitely many maximal vertex stabilizers up to conjugation. Then

(i) The closure D of D = {t ∈ T | Gt 6= 1} is a proﬁnite G-invariant subgraph of T having ﬁnitely many connected components Σi, i = 1, . . . , m, up to translation.

(ii) for the setwise stabilizer Gi = StabG(Σi) the quotient graph Gi\Σi has ﬁnite diameter and Σi contains a Gi-invariant subtree Di such that Gi\Di is ﬁnite. `m (iii) G = i=1 Gi q F is a free pro-p product, where F is a free pro-p group acting freely on T .

16 Proof. We follow the idea of the proof of [21, Theorem 3.5]. Since the action is n-acylindrical, T Gt has diameter at most n for every non-trivial edge or vertex stabilizer G . Note that D = S T Gt . We first show that G\D has finite diameter t Gt6=1 (as an abstract graph). Indeed, since there are only finitely many maximal vertex stabilizers

up to conjugation, say Gv1 ,...Gvk , it suffices to show that for a maximal vertex stabilizer Gvi , S Gt the tree T has finite diameter (if non-empty). But for 1 6= Gt ≤ Gvi the geodesic 16=Gt≤Gvi [t, vi] is stabilized by Gt (cf. [17, Corollary 3.8]) and so has length at most n. Thus G\D as an abstract graph has finite diameter (at most 2nk) and finitely many connected components (at most k). It follows that the closure ∆ of G\D in G\T has also finitely many (profinite) connected components (at most k) and finite diameter (at most 2nk), see Lemma 3.2. Note that the preimage of ∆ in T is exactly D. Since ∆ has finite diameter and so all its connected components are connected as abstract graphs by [8, Corollary 4], it is immediate that connected components of D are mapped surjectively onto corresponding connected components of ∆ (Indeed, if Σ is a connected component of D and Ω is its image in ∆, let e be an edge with say d0(e) ∈ Ω. Then 0 0 0 for an edge e ∈ D which is mapped onto e there exists g ∈ G with d0(ge ) ∈ Σ so that ge ∈ Σ and so e ∈ Ω). Thus the number of connected components of D up to translation equals the number of connected components of ∆ (≤ k). This proves (i). Collapsing all connected components of D, by Proposition on page 486 of [28] or [13, Propo- sition 3.9.1, as well as Cor. 3.10.2 and Prop. 3.10.4(b)], we get a pro-p tree T¯ on which G acts with trivial edge stabilizers, and vertex stabilizers being the setwise stabilizers of the connected ¯ 0 0 components of D. In particular, we have only finitely many vertices v1, . . . , vm up to translation whose stabilizers G 0 are non-trivial (and m ≤ k). So by Theorem 2.21, G is a free pro-p vi product m a G = G 0 q F, vi i=1 where F is naturally isomorphic to G/G˜ with G˜ taken with respect to the action on T¯, i.e. G G = hG 0 ,...,G 0 i . Then F acts freely on T and (iii) is proved. e v1 vm

By [2, Lemma 2.14], for any connected component Σi of D and its setwise stabilizer Gi = StabG(Σi) we have Gi\Σi ⊆ ∆ and so Gi\Σi is a connected component ∆i of ∆. So ∆i has ﬁnite diameter and by Proposition 3.5 Σi possesses a Gi-invariant subtree Di such that Gi\Di is ﬁnite. This proves (ii).

Corollary 5.5. Let G be a finitely generated pro-p group which is the fundamental group of a finite graph (G, Γ) of pro-p groups, and let T = T (G) its standard pro-p tree. Let H be a finitely generated subgroup of G that acts n-acylindrically on T , with finitely many maximal `m vertex stabilizers up to conjugation. Then H = i=1 Hi q F (possibly with one factor) with F free pro-p and there exists an open subgroup U of G containing H such that

(i) The natural map F −→ U/U˜ is injective.

17 (ii) U = Π1(U,U\T ), Hi = Π1(Hi,Hi\Di) (where Di is a minimal Hi-invariant subtree of T ) and (Hi,Hi\Di) are disjoint subgraphs of groups of (U,U\T ).

Moreover, the latter statements (i) and (ii) hold for any open subgroup V of U containing the group H.

Proof. By Theorem 5.4 applied to the action of H on T , there are subgroups Hi (i = 1, . . . , m) `m and F of H, with F free pro-p, such that H = i=1 Hi qF . Furthermore there are Hi-invariant H subtrees Di of T with Hi\Di finite, and Hi = StabH (Di), and F = H/hH1,...,Hmi . Note ¯ that the Hi\Di are disjoint subgraphs of H\T , and are contained in D (notation as in Theorem S 5.4). Choose an open subgroup U of G containing H such that the map i(Hi\Di) → U\T ˜ S is injective and the map F → U/U is injective (this is possible since i Hi\Di is finite, F is ˜ finitely generated and U/U is free pro-p). Then the Hi\Di are disjoint in U\T , whence if we choose maximal subtrees Ti of Hi\Di, their union extends to a maximal subtree of U\T . As G\T is finite, so is U\T , and we can then apply (the proof of) Lemma 3.7 to get the result.

6 Generalized accessible pro-p groups

We apply here the results of the previous section to finitely generated generalized accessible pro-p groups in the sense of the following definition. Definition 6.1. Let F be a family of pro-p groups. A pro-p group G will be called F-accessible if there is a number n = n(G) such that any finite, proper, reduced graph of pro-p groups with edge groups in F having fundamental group isomorphic to G has at most n edges. The definition generalizes the definition of accessibility given in [24], where the edge groups are finite. In fact if F is the class of all finite p-groups, an F-accessible pro-p group will simply be called accessible. Proposition 6.2. Let F be a family of pro-p groups and G a finitely generated F-accessible pro-p group acting on a pro-p tree T with edge stabilizers in F. Then G has only finitely many maximal vertex stabilizers up to conjugation, and in fact the number of such stabilizers does not exceed n(G) + 1.

Proof. Let Gv1 ,...Gvm be maximal vertex stabilizers which are non-conjugate. We will show ˜ ˜ ˜ that m is bounded. If U/0 G, then G/U acts on U\T and by Lemma 4.1, G = lim G/U, ←−U/oG ˜ where G/U = Π1(GU , ΓU ) is the fundamental group of a finite reduced graph of finite p-groups. ˜ ˜ ˜ ˜ Thus starting from a certain U the stabilizers Gv1 U/U,...Gvm U/U of the images of v1, . . . , vm in ˜ ˜ U\T are still maximal and distinct. So they are maximal finite subgroups of G/U = Π1(GU , ΓU ) and so are conjugate to vertex groups of (GU , ΓU ) (see Theorem 2.22). Therefore ΓU has at least m vertices. Then by Corollary 4.4, G admits a decomposition as the fundamental group of a reduced finite graph of pro-p groups Π1(G, ΓU ) with edge groups in F and so m ≤ n(G) + 1.

18 Theorem 6.3. Let G be a ﬁnitely generated F-accessible pro-p group acting on a pro-p tree T with edge stabilizers in F. Then G is the fundamental pro-p group of a ﬁnite graph of pro-p groups (G, Γ), where each vertex group G(v) and each edge group G(e) is a vertex stabilizer Gv˜ and an edge stabilizer Ge˜ respectively (for some v,˜ e˜ ∈ T ). Moreover, the size of V (Γ) is bounded by n(G) + 1 for every such (G, Γ).

Proof. By Proposition 6.2 the number of maximal vertex stabilizers of G is bounded by n(G)+1. Therefore the result follows from Theorem 5.1 and Corollary 5.2.

Remark 6.4. In the classical Bass-Serre theory of groups acting on trees, structure theorems like Theorem 6.3 are used to obtain structure results on subgroups of fundamental groups of graphs of groups (see for example [19, §5]). In our situation, to use Theorem 6.3 for this purpose one needs to assume that F is closed under subgroups. A relevant to the context such general example is the class of small pro-p groups. Namely, one can follow the approach of [1] in the abstract case and call a pro-p group G small if whenever G acts on a pro-p-tree T , and K ≤ G acts freely on T , then K is procyclic. If the action on T is associated with splitting G into a free amalgamated product G = G1 qH G2 or an HNN-extension G = HNN(G1, H, t), it means that H is normal in G with G/H either procyclic or infinite dihedral (see [17, Theorem 3.15]). The class of small pro-p groups S is closed under subgroups. Then one can use Theorem 6.3 to prove the following statement. Let G be a pro-p group acting on a pro-p tree T with small edge stabilizers. Let H be a finitely generated S-accessible subgroup of G. Then H = Π1(H, Γ) is the fundamental group of a finite graph of pro-p groups (H, Γ), where each vertex group H(v) and each edge group H(e) is a vertex stabilizer Hv˜ and an edge stabilizer He˜ respectively (for some v,˜ e˜ ∈ T ). Moreover, the cardinality of E(Γ) is bounded by the accessibility number n(H). Note also that a finitely generated free pro-p group F is S-accessible, since a free small pro-p group has to be pro-cyclic.

Example 6.5. If C is a class of pro-cyclic pro-p groups then any ﬁnitely generated pro-p group is C-accessible ([21, Lemma 3.2]).

In fact we can bound the C-accessibility number n(G) in terms of the minimal number of generators d(G) of G.

Theorem 6.6. Let G = π1(Γ, G) be the fundamental group of a ﬁnite graph of pro-p-groups, with procyclic edge groups, and assume that d(G) = d ≥ 2. Then (assuming that the graph is reduced), the vertex groups are ﬁnitely generated, the number of vertices of Γ is ≤ 2d − 1, and the number of edges of Γ is ≤ 3d − 2.

Proof. Let T be a maximal subtree of Γ and H = Π1(G,T ) be the fundamental group of the tree of groups (G,T ) obtained by restricting (G, Γ) to T . Then G = HNN(H,C1,...,C`, t1, . . . , t`),

19 where ` = |E(Γ)| − |E(T )|. Note that the quotient of G by the normal subgroup generated by H is free on t1, . . . , t`, so d(G) ≥ `. Since |V (Γ)| = |V (T )|, and |E(T )| = |V (T )| − 1 we have |E(Γ)| = |V (T )| − 1 + ` ≤ |V (T )| + d − 1. It therefore suﬃces to show that |V (T )| = |V (Γ)| ≤ 2d − 1. Consider the Mayer-Vietoris sequence

→ ⊕e∈E(T )H1(G(e)) → ⊕v∈V (T )H1(G(v)) → H1(G) → Fp[[E(T )]] → Fp[[V (T )]] → Fp → 0 (see [13, Thm 9.4.1]). Since T is a tree,

0 → Fp[[E(T )]] → Fp[[V (T )]] → Fp → 0

is exact (see Subsection 2.2) and so H1(G) → Fp[[E(T )]] is the zero map, and ⊕v∈V (T )H1(G(v)) → H1(G) is onto. Let n be the number of vertices of T , let m be the number of vertices whose vertex groups are cyclic and k be the number of vertices whose vertex groups are not cyclic, so that n = m+k and the number of edges of T is n − 1. Then

d(G) = dim(H1(G)) ≥ m + 2k − (n − 1) = m + 2k − n + 1 = k + 1

(here we are using that all edge groups are cyclic). On the other hand if the vertex group G(v) is cyclic and e is incident to v then the natural map H1(G(e)) → H1(G(v)) is the zero map (because G(e) ≤ Φ(G(v))). Denoting by Vc the set of vertices with cyclic vertex group,

it follows that ⊕v∈Vc H1(G(v)) intersects trivially the image of ⊕e∈E(T )H1(G(e)) and therefore maps injectively into H1(G). Therefore m ≤ d. Thus n = k +m ≤ d(G)−1+d(G) = 2d(G)−1 as required. Finally, since ⊕e∈E(T )H1(G(e)) and H1(G) are ﬁnite, so is ⊕v∈V (T )H1(G(v)), i.e. G(v) is ﬁnitely generated for every v.

We now apply Theorem 6.6 to the pro-p analogue of a limit groups defined in [10]. It is worth recalling their definition. Denote by G0 the class of all free pro-p groups of finite rank. We define inductively the class Gn of pro-p groups Gn in the following way: Gn is a free amalgamated pro-p product Gn−1 qC A, where Gn−1 is any group from the class Gn−1, C is any self-centralizing procyclic pro-p subgroup of Gn−1 and A is any finite rank free abelian pro-p group such that C is a direct summand of A. The class of pro-p groups L (pro-p limit groups) consists of all finitely generated pro-p subgroups H of some Gn ∈ Gn, where n ≥ 0. Then H is a subgroup of a free amalgamated ∼ ∼ m pro-p product Gn = Gn−1 qC A, where Gn−1 ∈ Gn−1, C = Zp and A = C × B = Zp . By Theorem 3.2 in [14], this amalgamated pro-p product is proper. Thus H acts naturally on the pro-p tree T associated to Gn and its edge stabilizers are procyclic. An immediate application of Theorem 6.6 then gives a bound on the C-accessibility number of a limit pro-p group.

20 Corollary 6.7. Let G be a pro-p limit group. Then G is the fundamental group of a finite graph of finitely generated pro-p groups (G, Γ), where each edge group G(e) is infinite procyclic. Moreover, |V (Γ)| ≤ 2d − 1, and |E(Γ)| ≤ 3d − 2, where d is the minimal number of generators of G.

Accessible pro-p groups

This subsection is dedicated to accessible pro-p groups. Note that there exists a finitely generated non-accessible pro-p group (see [24]) and that it is an open question of whether a finitely presented pro-p group is accessible. The next proposition gives a characterization of accessible pro-p groups. Proposition 6.8. Let G be a finitely generated pro-p group. Then G is accessible if and only if it is a virtually free pro-p product of finitely many virtually freely indecomposable pro-p groups. Proof. Let H be an open subgroup of G that splits as a free pro-p product of virtually freely indecomposable pro-p groups. Replacing H by the core of H in G and applying the Kurosh subgroup theorem for open subgroups (cf. [15, Thm. 9.1.9]), we may assume that H is normal in G. Refining the free decomposition if necessary and collecting free factors isomorphic to Zp we obtain a free decomposition

H = F q H1 q · · · q Hs, (3)

where F is a free subgroup of rank t, and the Hi are virtually freely indecomposable finitely generated subgroups which are not isomorphic to Zp. By [23, Theorem 3.6] G = Π1(G, Γ) is the fundamental pro-p group of a finite graph of pro-p groups with finite edge groups. Moreover, it follows from its proof (step 2) that H intersects all edge groups trivially. Then by [24, Theorem p[G:H] 3.1] Γ has at most p−1 (d(G) − 1) + 1 edges. So G is accessible. Conversely, suppose G is accessible. Write G = Π1(G, Γ), where (G, Γ) is a finite graph of pro-p groups with finite edge groups, and such that Γ is of maximal size. Choosing an open normal subgroup H intersecting all edge groups of G trivially we have a a H = H ∩ G(v)gv q F,

v∈V (Γ) gv∈H\G/G(v)

where gv runs through double cosets representatives of H\G/G(v) and F is free pro-p of ﬁnite rank (see Theorem 2.21 with the action of H on the standard pro-p tree T (G)). Since Γ is of maximal size, G(v) does not split as an amalgamated free pro-p product or HNN extension over a ﬁnite p-group, so by [23, Theorem A] G(v) is not a virtual free pro-p product, in particular gv H ∩ G(v) is freely indecomposable. Since F is a free pro-p product of Zp’s the result follows.

Question 6.9. Let G be a ﬁnitely generated pro-p group acting on a pro-p tree with ﬁnite vertex stabilizers. Is G accessible?

21 1 Note that H (G, Fp[[G]]) is a right Fp[[G]]-module. The next theorem gives a suﬃcient condition of accessibility for a pro-p group in terms of this module; we do not know whether the converse also holds (it holds in the abstract case).

1 Theorem 6.10. Let G be a ﬁnitely generated pro-p group. If H (G, Fp[[G]]) is a ﬁnitely generated Fp[[G]]-module, then G is accessible.

Proof. Suppose G = Π1(G, Γ) is the fundamental group of a reduced finite graph (G, Γ) of pro-p groups with finite edge groups. We will first do the case where if v is any vertex of Γ, then 1 G(v) is infinite and H (G(v), Fp[[G(v)]]) 6= 0. The group G acts on the standard pro-p tree T associated to (G, Γ), and we get

0 −→ ⊕e∈E(Γ)Fp[[G/G(e)]] −→ ⊕v∈V (Γ)Fp[[G/G(v)]] −→ Fp −→ 0

Applying HomFp[[G]](−, Fp[[G]]) to this exact sequence and taking into account that

G HomFp[[G]](Fp, Fp[[G]]) = (Fp[[G]]) = 0 ([11, Lemma 3]), we get

0 → ⊕v∈V (Γ)HomFp[[G]](Fp[[G/G(v)]], Fp[[G]]) → ⊕e∈E(Γ)HomFp[[G]](Fp[[G/G(e)]], Fp[[G]]) → H1(G, [[G]]) → ⊕ Ext1 ( [[G/G(v)]], [[G]]) → ⊕ Ext1 ( [[G/G(e)]], [[G]]) Fp v∈V (Γ) Fp[[G]] Fp Fp e∈E(Γ) Fp[[G]] Fp Fp By Shapiro’s lemma

0 G(v) HomFp[[G]](Fp[[G/G(v)]], Fp[[G]]) = H (G(v), ResFp[[G(v)]]Fp[[G]]) = (Fp[[G]]) = 0, the latter equality since G(v) is inﬁnite ([11, Lemma 3]); similarly, by Shapiro’s lemma,

Ext1 ( [[G/G(v)]], [[G]]) = H1(G(v), Res [[G]]), Fp[[G]] Fp Fp Fp[[G(v)]]Fp

0 HomFp[[G]](Fp[[G/G(e)]], Fp[[G]]) = H (G(e), ResFp[G(e)]Fp[[G]]), Ext1 ( [[G/G(e)]], [[G]]) = H1(G(e), Res [[G]]). Fp[[G]] Fp Fp Fp[G(e)]Fp Now since G(e) is ﬁnite, G has a system of open normal subgroups U intersecting G(e) trivially and so M Res [[G]] = lim Res [[G/U]] = lim( [G(e)]). Fp[G(e)]Fp ←− Fp[G(e)]Fp ←− Fp U U G(e)\G/U

Moreover, since Hom commutes with projective limits in the second variable we have M H0(G(e), Res [[G]]) = lim( H0(G(e), [G(e)])). Fp[G(e)]Fp ←− Fp U G(e)\G/U

22 0 ∼ But H (G(e), Fp[G(e)]) = Fp. Thus M M lim( H0(G(e), [G(e)])) = lim( ) = lim [[G(e)\G/U]]) = [[G(e)\G]]. ←− Fp ←− Fp ←− Fp Fp U G(e)\G/U U G(e)\G/U U

Since Fp[[G]] is a free Fp[G(e)] and Fp[[G(v)]]-module, (see [26, Proposition 7.6.3]) and so is projective by [26, Proposition 7.6.2], and since Fp[[G]] is a local ring, by [26, Proposition 7.5.1] and [26, Proposition 7.4.1 (b)] Y ResFp[G(e)]Fp[[G]]) = Fp[G(e)] I and Y ResFp[[G(v)]]Fp[[G]]) = Fp[[G(v)]] j∈J for some inﬁnite sets of indices I,J. 1 Note also that Ext commutes with direct products on the second variable and H (G(e), Fp[G(e)]) = 0, since a free Fp[G(e)]-module is injective. So

1 Y 1 H (G(e), ResFp[G(e)]Fp[[G]]) = H (G(e), Fp[G(e)])) = 0 I Thus the above long exact sequence can be rewritten as

1 1 0 −→ ⊕e∈E(Γ)Fp[[G/G(e)]] −→ H (G, Fp[[G]]) −→ ⊕v∈V (Γ)H (G(v), ResFp[[G(v)]]Fp[[G]]) −→ 0

1 We show now that H (G(v), ResFp[[G(v)]]Fp[[G]]) 6= 0 for each v. Indeed, since Ext com- 1 mutes with the direct product on the second variable we have H (G(v), ResFp[[G(v)]]Fp[[G]]) = Q 1 1 j∈J H (G(v), Fp[[G(v)]]). But for every v the groups H (G(v), Fp[[G(v)]]) 6= 0 by the assump- 1 tion at the beginning of the proof. So H (G(v), ResFp[[G(v)]]Fp[[G]]) 6= 0 for every v. Hence the number of vertices in Γ cannot exceed the minimal number of generators of 1 Fp[[G]]-module H (G, Fp[[G]]). The number of edges of Γ cannot exceed d(G)+|V (Γ)|−1 since ˜ ˜ the rank of π1(Γ) = G/G equals |E(Γ)| − |V (Γ)| + 1, where the equality π1(Γ) = G/G follows from [13, Corollary 3.9.3] combined with [13, Proposition 3.10.4 (b)].

We will now do the general case. First observe that if G = Π1(G, Γ), where (G, Γ) is a reduced finite graph of pro-p-groups with finite edge groups, then, letting T be a maximal subtree of Γ, there are at most d := d(G) edges in Γ \ T , and therefore there are at most 3d pending vertices in Γ, see Lemma 3.6. We will now bound the size of T . 1 Suppose now that some vertex group G(v) is either finite or has H (G(v), Fp[[G(v)]]) = 0. If e ∈ E(T ) is adjacent to v, with other extremity w, then collapsing {e, v, w} into a new vertex y, and putting on top of y the group G(y) = G(v) qG(e) G(w), by Theorem 4.5 we 1 have H (G(y), Fp[[G(y)]]) 6= 0, and G(y) is infinite. Let M be the number of generators of 1 H (G, Fp[[G]]).

23 Claim. The diameter of T is at most 2M. Indeed, if not, it contains a path with 2M + 2 distinct vertices. But applying the above procedure to get rid of the bad vertices on the path, produces at least M + 1 vertices y with 1 H (G(y), Fp[[G(y)]]) 6= 0 and G(y) inﬁnite, which contradicts the ﬁrst part.

The result now follows, as there is a bound on the size of trees with at most 3d pending vertices and diameter ≤ 2M, and |Γ \ T | ≤ d (see Proposition 3.4).

7 Howson’s property

Definition 7.1. We say that a pro-p group G has Howson’s property, or is Howson, if whenever H and K are two finitely generated closed subgroups of G, then H ∩ K is finitely generated. Free and Demushkin pro-p-groups are Howson [12, 20] and the Howson property is preserved under free (pro-p) products, see [20, Thm 1.9]. In this section we investigate the preservation of Howson’s property under various (free) constructions. Lemma 7.2. Let G be a finitely generated pro-p group acting on a profinite space Y such that the number of maximal point stabilizers Ga, up to conjugation, is finite and represented by the elements a ∈ A ⊆ Y . Let H be a subgroup of G such that Hy is finitely generated for each y ∈ Y and the kernel of the natural homomorphism β : Fp[[H\Y ]] −→ Fp[[G\Y ]] is finite. Then the image of the natural homomorphism η : H1(H, Fp[[Y ]]) −→ H1(H) is finite.

Proof. We use the characterisation H1(G) = G/Φ(G). By Shapiro’s lemma H1(G, Fp[[G/Gy]]) = H1(Gy) = Gy/Φ(Gy) and so the image of γ : H1(G, Fp[[Y ]]) −→ H1(G) coincides with the smallest closed subgroup containing all images of the H1(Gy)’s. Observe now that if Gy ≤ Ga and g g ∈ G, then GyΦ(G) ≤ GaΦ(G) = GaΦ(G), whence the image of H1(Gy) in H1(G) is contained g in the image of H1(Ga), which equals the image of H1(Ga). Thus the image of γ coincides with the subgroup of H1(G) generated by the images of H1(Ga), a ∈ A. A given H-orbit H/Hy ⊂ Y is sent by β to a subset of a G-orbit G/Gy in Y . If for y ∈ Y the stabilizer Gy is not maximal, then there exists a maximal Ga, a ∈ A, and g ∈ G, such that g g Gy ≤ Ga. Hence Hy ≤ Ha . Since ker(β) is finite, the set B of H-orbits in Y that map into some G-orbit Ga with a ∈ A, is finite. Note that H1(H, Fp[[H/Ha]]) = H1(Ha) (by Shapiro’s lemma). Thus the image of η : H1(H, Fp[[Y ]]) −→ H1(H) coincides with the group generated g by the images of H1(Hb), b ∈ B. But Ha is finitely generated, so each H1(Ha ) is finite, and since B is finite, the image of η is also finite.

Theorem 7.3. Let G be a finitely generated pro-p group acting on a pro-p tree T such that G\T is finite. Let H be a finitely generated subgroup of G such that H\T (H) is finite, where T (H) is a minimal H-invariant subtree of T , and [Ge : He] < ∞ for all e ∈ E(T (H)). If K is a finitely generated subgroup of G, then H ∩ K is finitely generated in each of the following cases:

24 (i) K intersects trivially all vertex stabilizers Hv, v ∈ V (T (H)); (ii) K acts on T with ﬁnitely many maximal vertex stabilizers up to conjugation, the vertex stabilizers Hv,Kv are ﬁnitely generated and the Gv are Howson, v ∈ V (T ).

Proof. The proof follows the idea of the proof of Theorem 1.9 in [20]. Put ∆ = H\T (H). Observe that if u ∈ T (H), then Hu is the intersection of all Uu with U an open subgroup of G containing H. As ∆ is finite, there is some open subgroup U of G containing H and such that we have an injection H\T (H) → U\T . By Lemma 3.7, passing to an open subgroup of G containing H, we may therefore assume that G = Π1(G, Γ) with Γ finite, and that (H, ∆) is a subgraph of groups of (G, Γ) such that H = Π1(H, ∆). Note that then [G(e): H(e)] < ∞ for all e ∈ E(∆), and, since ∆ is finite, replacing G by an open subgroup containing H, we may assume that G(e) = H(e) for every e ∈ E(∆). Let 0 −→ Fp[[E(T )]] −→ Fp[[V (T )]] −→ Fp −→ 0 and 0 −→ Fp[[E(T (H))]] −→ Fp[[V (T (H))]] −→ Fp −→ 0 be the short exact sequences associated with T and T (H) (note that E(T ), E(T (H)) are compact since G\T is finite). ˆ ˆ Applying to them Fp⊗Fp[[K]] and Fp⊗Fp[[K∩H]] respectively we get the following long exact sequences (see [13, (9.6), page 265]):

ˆ H1(K, Fp[[V (T )]]) → H1(K, Fp) → Fp⊗Fp[[K]]Fp[[E(T )]] → δ ˆ −→ Fp⊗Fp[[K]]Fp[[V (T )]] → Fp → 0 (4) and

H1(K ∩ H, Fp[[V (T (H))]]) → H1(H ∩ K, Fp)→ ˆ σ ˆ Fp⊗Fp[[K]]Fp[[E(T (H))]] −→ Fp⊗Fp[[K]]Fp[[V (T )]] → Fp → 0. (5) Using the deﬁnitions of T and T (H) (see 2.17), we can rewrite the long exact sequences as follows:

⊕ H1(K, Fp[[G/G(v)]]) → H1(K, Fp) → ⊕ Fp[[K\G/G(e)]] → v∈G\V (T ) e∈G\E(T ) δ −→ ⊕ Fp[[K\G/G(v)]] → Fp → 0 (6) v∈G\V (T ) and

25 M H1(K ∩ H, Fp[[H/H(v)]]) → H1(H ∩ K, Fp)→ v∈H\V (T (H)) M σ M Fp[[K ∩ H\H/H(e)]] −→ Fp[[(K ∩ H)\H/H(v)]] → Fp → 0. (7) e∈H\E(T (H)) v∈H\V (T (H)) We then have the following commutative diagramme:

M M Fp[[K\G/G(e)]] δ - Fp[[K\G/G(v)]] e∈G\E(T ) v∈G\V (T ) 6 6

α β

M M Fp[[K ∩ H\H/H(e)]] σ - Fp[[K ∩ H\H/H(v)]] e∈H\E(T (H)) v∈H\V (T (H))

We want to show that ker β ◦σ is finite, or equivalently that ker δ◦α is finite. The dimension (as an Fp-v.s) of ker δ is ≤ dim(H1(K)), i.e., less than or equal to the number of generators of K. So, we need to show that ker(α) is finite, and if possible bound its size. We know that there is an inclusion of (K ∩ H)\H in K\G, and we need to see what happens when we quotient by the action of G(e) (on the right).

The inclusion map Fp[[(K ∩ H)\H]] → Fp[[K\G]] is a map of right Fp[[He]]-modules for any e ∈ E(T (H)), and note that it sends distinct He-orbits to distinct Ge-orbits (because G(e) = H(e) for e ∈ E(∆) and so Ge = He for every e ∈ E(T (H))). Hence α is an injection!!

To summarise: δ◦α = β ◦σ have ﬁnite kernel, of dimension bounded by d(K). Furthermore, L as the image of σ in v∈H\V (T (H)) Fp[[K∩H\H/H(v)]] has codimension 1 (by the exact sequence (7)), it follows that ker(β) is also ﬁnite, and we have dim(ker(σ)) + dim(ker(β)) ≤ d(K) + 1.

(i) if K intersects trivially all conjugates of Hv then the left term of (7) is M H1(K ∩ H, Fp[[H/H(v)]]) v∈H\V (T (H)) L and equals 0 because v∈H\V (T (H)) Fp[[H/H(v)]] is a free K ∩ H-module, so (i) follows from the injectivity of α.

(ii) Since the G(v), v ∈ V (∆), are Howson, and Kv,Hv are finitely generated, Kv ∩ Hv is finitely generated for any v ∈ V (T ). Since ker(β) is finite, we can apply Lemma 7.2 to

26 L K ∩ H ≤ K to deduce that the image of H1(K ∩ H, v∈H\V (T (H)) Fp[[H/H(v)]]) in H1(H ∩ K) is finite. Combining this with the finiteness of ker(σ) we deduce that H1(H ∩ K) is finite, i.e. H ∩ K is finitely generated. Corollary 7.4. Let G be a finitely generated pro-p group acting on a pro-p tree T with procyclic edge stabilizers and Howson’s vertex stabilizers. Let H be a finitely generated subgroup of G that does not split as a free pro-p product and whose action on T is n-acylindrical. If K is a finitely generated subgroup of G then H ∩ K is finitely generated. Proof. If H is procyclic, there is nothing to prove. Otherwise, by Corollary 5.5 there exists a minimal H-invariant subtree D of T with ∆ = H\D finite and H = Π1(H, ∆). Moreover, since H is not a non-trivial free pro-p product, H(e) 6= 1 for all e ∈ E(∆) (indeed, if H(e) = 1 then as explained in the proof of Theorem 4.2 either G = G1 qG2, where G1,G2 are the fundamental groups of graphs of groups restricted to the connected components of ∆ \{e}, or G = G1 q Zp, where G1 is the fundamental groups of the graph of groups restricted to ∆ \{e}). Hence for each e ∈ E(∆) we have [G(e): H(e)] < ∞. Vertex stabilizers Kv,Hv are finitely generated by Theorem 6.6, and by Example 6.5 and Corollary 6.2 the set of maximal vertex K-stabilizers is finite up to conjugation. Thus all hypotheses of Theorem 7.3 are satisfied from which we deduce the result.

Corollary 7.5. Let G = G1 qC G2 be a free amalgamated pro-p product of free or Demushkin not soluble pro-p groups with C maximal procyclic in G1 or G2. Let H be a finitely generated subgroup of G that does not split as a free pro-p product. Then H ∩ K is finitely generated for any finitely generated subgroup K of G. Proof. In this case the action of G on its standard pro-p tree T is 2-acylindrical. This follows g1 from the fact that if C is maximal procyclic in say G1, then C is malnormal in G1, i.e. C ∩C = g1 g1 1 for any g1 ∈ G1 \ C. Indeed, if C ∩ C 6= 1 then hC,C i normalizes this intersection. But every 2-generated subgroup of G1 is free and so can not have procylic normal subgroups, so g1 C = C and so g1 normalizes C. But then the same applies to hC, g1i so it is procyclic contradicting the maximality of C in G1. Thus using that free and Demushkin pro-p groups are Howson [12, 20], by Corollary 7.4 we obtain the result.

Theorem 7.6. Let G = G1 qC G2 be a free pro-p product with procyclic amalgamation. Let Hi ≤ Gi, be finitely generated such that C ∩ H1 ∩ H2 6= 1, H = hH1,H2i and K ≤ G a finitely generated subgroup of G. Then K ∩ H is finitely generated in each of the following cases

(i) K intersects all conjugates of the Hi trivially. Moreover, if C ≤ Hi (i = 1, 2) then d(H ∩ K) ≤ d(K).

27 (ii) The Gi’s are Howson pro-p. Proof. The group G acts on its standard pro-p tree T = T (G) and so we can apply Theorem 7.3 to deduce the first part of (i). Vertex stabilizers Kv,Hv are finitely generated by Theorem 6.6, and by Example 6.5 and Corollary 6.2 the set of maximal vertex K-stabilizers is finite up to conjugation. Thus all hypotheses of Theorem 7.3 are satisfied to deduce (ii). Thus, we only need to show the second part of (i). The proof of Theorem 7.3 involved replacing G by a subgroup of finite index, and so to obtain the precise bound d(K), we need to show that this is not necessary. Towards that, our hypothesis C ≤ H1 ∩ H2 implies that, if T (H) is a minimal H-invariant subtree of T , then H\T (H) ' G\T and G(e) = H(e). As was observed in the proof of Theorem 7.3, δ ◦ α = β ◦ σ have finite dimensional kernel, of dimension bounded by d(K). If K does not intersect the conjugates of the Hi then the left term of equation (7) (in the proof of 7.3) is 0, so from the injectivity of α one deduces that the natural map H1(K ∩H) −→ H1(K) is an injection.

Theorem 7.7. Let G = HNN(G1, C, t) be a pro-p HNN extension, G1 a finitely generated and C 6= 1 procyclic. Let H1 be a finitely generated subgroup of G1 such that C ∩ H1 6= 1 and H = hH1, ti. Then for a finitely generated subgroup K of G the intersection K ∩ H is finitely generated in each of the following cases

(i) K intersects trivially every conjugate of H1. Moreover, if C ≤ H1 then d(H ∩K) ≤ d(K).

(ii) G1 satisfies Howson’s property. Proof. The proof is identical to the proof of Theorem 7.6. The group G acts on its standard pro-p tree T = T (G) and so we can apply Theorem 7.3 to deduce the first part of (i). Vertex stabilizers Kv,Hv are finitely generated by Theorem 6.6, and by Example 6.5 and Corollary 6.2 the set of maximal vertex K-stabilizers is finite up to conjugation. Thus all hypotheses of Theorem 7.3 are satisfied to deduce (ii). Thus, we only need to show the second part of (i), and this is done exactly as in Theorem 7.6.

8 Normalizers

Proposition 8.1. Let C be a procyclic pro-p group and U ≤ C a subgroup of C.

(a) Let G = G1 qC G2 and N = NG(U). Then N = NG1 (C) qC NG2 (C).

(b) Let G = HNN(G1, C, t) be a proper pro-p HNN-extension and N = NG(U). g t 0 (i) If there is some g ∈ G1 such that U = U , then N = HNN(NG1 (U), C, t ). t (ii) If U and U are not conjugate in G1 then N = N1 qC N2, where N1 = N t−1 (U) and G1

N2 = NG1 (U).

28 Proof. The proof follows the strategy of proofs in the ordinary profinite case of Proposition 2.5 in [16] or Proposition 15.2.4 (b) [13]. Let T be the standard pro-p tree for G and recall that E(T ) = G/C, V (T ) = G/G1 ∪· G/G2 in Case (a) and V (T ) = G/G1 in Case (b). By [17, Theorem 3.7] the subset Y = T U of T consisting of points fixed by U is a pro-p subtree. Observe that if g ∈ G, then U fixes gC if and only if U g ≤ C.

Then N acts on Y continuously. Indeed, if g ∈ N, y ∈ Y and u ∈ U, then ug = gu0 for some u0 ∈ U, and therefore ugy = gu0y = gy. This being true for all u in U, we get that N acts on Y .

Consider the natural epimorphism ϕ : T → G\T . Then the natural map ψ : Y → N\Y is the restriction of ϕ to Y . To see this pick h ∈ G such that hC ∈ E(Y ); so U h ≤ C and therefore U, U h ≤ C. As C is procyclic, we get U = U h, i.e., h ∈ N (work in finite quotients of G where the equality is obvious). This shows that ψ coincides with the restriction of ϕ to Y . Thus N\E(Y ) consists of one edge only, and therefore N\Y has at most two vertices. According to Proposition 4.4 in [30] (or [13, Theorem 6.6.1]), we have N = N1 qC N2, where 0 N1 = NG1 (U) and N2 = NG2 (U), or N = HNN(NG1 (U), C, t ), depending on whether Y has two vertices or just one vertex. (Note that N contains C). In Case (a) ϕ(gG1) 6= ϕ(gG2), so ψ(Y ) has two vertices. In Case (b) N\Y has one vertex only iff d1(C) = tG1 is in the N-orbit of d0(C) = G1, i.e., if t n nt−1 −1 g t G1 = G1 for some n ∈ N iff G1 = G1 iff g = nt ∈ G1, in which case U = U as required.

Proposition 8.2. Let G be a pro-p group acting on a pro-p tree T and U be a procyclic subgroup of G that does not stabilize any edge. Then one of the following happens:

(1) For some vertex v, U ≤ Gv: then NG(U) = NGv (U).

(2) For all vertices v, U ∩ Gv = {1}. Then NG(U)/K is either isomorphic to Zp or to a

generalized dihedral group Z2 q2Z2 Z2 = Z2 o Z2, where K is some normal subgroup of NG(U) contained in the stabilizer of an edge.

Proof. Let N = NG(U) and let D be a minimal U-invariant subtree of T (that exists by Proposition 2.19). Case 1. |D| = 1, i.e., U stabilizes a vertex v. If v 6= nv for some n ∈ N, then by [17, Corollary 3.8], U stabilizes all edges in [v, nv], contradicting our hypothesis. So NG(U) ﬁxes v and we have (1). Case 2. D is not a vertex. Then U acts irreducibly on D and so by Proposition 2.19 it is unique. Note that if n ∈ N, then nD is also D-invariant, and therefore must equal D. Hence N acts irreducibly on D and by Lemma 4.2.6(c) in [13], CN (U)K/K is free pro-p, where K is the kernel of the action (and is the intersection of all stabilizers). Hence CN (U)K/K is procyclic ∼ (because UK/K is procyclic, 6= 1) and so N/K is either Zp or C2 qC2, since Aut(U) = Zp ×Cp−1 for p > 2 or Z2 × C2 for p = 2.

29 Combining Theorem 7.6 and Propositions 8.1 and 8.2 we deduce the following

Theorem 8.3. Let C be a procyclic pro-p group and G = G1 qC G2 be a free amalgamated pro-p product or a pro-p HNN-extension G = HNN(G1, C, t) of Howson groups. Let U be a g procyclic subgroup of G and N = NG(U). Assume that NGi (U ) is finitely generated whenever g U ≤ Gi. If K ≤ G is finitely generated, then so is K ∩ N. Proof. Let T be the standard pro-p tree for G. If U does not stabilize any edge then by Proposition 8.2 either N (U) = N g (U) for some i and g, and so K ∩ N = K ∩ N g (U) is G Gi Gi finitely generated, or NG(U) is metacyclic and therefore so is K ∩ N. If U stabilizes an edge then U ≤ Cg for some g ∈ G and so we may assume that U ≤ C. Then by Proposition 8.1 N satisfies the hypothesis of either Theorem 7.6 or Theorem 7.7 and so by one of these theorems N ∩ K is finitely generated.

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P. A. Zalesskii Department of Mathematics University of Brasilia 70.910 Brasilia DF Brazil [email protected]